Download Lim.B.2

NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Mathematical Practices
A.1
Reasoning with definitions and theorems
MPAC 1
Students will be able to…
 Use definitions and theorems to build arguments, to justify conclusions or answers, and
to prove results.
 Confirm that hypotheses have been satisfied in order to apply the conclusion of a
theorem.
 Apply definitions and theorems in the process of solving a problem.
 Develop conjectures based on exploration with technology.
 Produce examples and counterexamples to clarify understanding of definitions, to
investigate whether converses of theorems are true or false, or to test conjectures.
A.2
Connecting concepts
MPAC 2
Students will be able to…
 Relate the concept of a limit to all aspects of calculus.
 Use the connection between concepts (e.g. rate of change and accumulation) or
processes (e.g. differentiation and its inverse process, antidifferentiation) to solve
problems.
 Connect concepts to their visual representations with and without technology.
 Identify a common underlying structure in problems involving different contextual
situations.
A.3
Implementing algebraic/computational processes
MPAC 3
Students will be able to…
 Select appropriate mathematical strategies.
 Sequence algebraic/computational procedures logically.
 Complete algebraic/computational processes correctly.
 Apply technology strategically to solve problems.
 Attend to precision graphically, numerically, analytically, and verbally and specify units of
measure.
 Connect the results of algebraic/computational process to the question asked.
A.4
Connecting multiple representations
MPAC 4
Students will be able to…
 Associate tables, graphs and symbolic representations of functions.
 Develop concepts using graphical, symbolical, or numerical representations with and
without technology.
 Identify how mathematical characteristics of functions are related in different
representations.
 Extract and interpret mathematical content from any presentation of a function (e.g.
utilize information from a table of values.)
 Construct one representational form from another (e.g. a table from a graph, etc.)
 Consider multiple representations of a function to select or construct a useful
representation for solving a problem
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
A.5
Building notational fluency
MPAC 3
Students will be able to…
𝑑𝑦
 Know and use a variety of notations (e.g. 𝑓’(𝑥), 𝑦 ′ , 𝑑𝑥 ).



Connect notation to definitions (e.g. relating the notation for the definite integral to that
of the limit of a Riemann sums).
Connect notation to different representations (graphical, numerical, analytical, and
verbal).
Assign meaning to notation, accurately interpreting the notation in a given problem and
across different contexts.
A.6
Communicating
MPAC 6
Students will be able to…
 Clearly present methods, reasoning, justifications, and conclusions.
 Use accurate and precise language and notation.
 Explain the meaning of expressions, notation, and results in terms of a context (including
units).
 Explain the connections among concepts.
 Critically interpret and accurately report information provided by technology.
 Analyze, evaluate, and compare the reasoning of others.
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Limits
Students will understand that:


The concept of a limit can be used to understand the behavior of functions
Continuity is a key property of functions that is defined using limits
Lim.B.1 Analyze functions for intervals of continuity or points of discontinuity
LO1.2A
Students will know that…
 A function 𝑓 is continuous at 𝑥 = 𝑐 provided that 𝑓(𝑐) exists, lim 𝑓(𝑥) exists, and
𝑥→𝑐
𝑓(𝑐) = lim 𝑓(𝑥).


𝑥→𝑐
Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are
continuous at all points in their domains.
Types of discontinuities include removable discontinuities, jump discontinuities, and
discontinuities due to vertical asymptotes.
Lim.B.2 Determine the applicability of important calculus theorems using continuity
LO1.2B
Students will know that…
 Continuity is an essential condition for theorems such as the Intermediate Value
Theorem, the Extreme Value Theorem, and the Mean Value Theorem.
Lim.B.3 Deduce and interpret behavior of functions using limits
LO1.1D
Students will know that…
 Asymptotic and unbounded behavior of functions can be explained using limits.
 Relative magnitudes of functions and their rates of change can be compared using limits
Lim.C.1 Express limits symbolically using correct notation
LO1.1Aa
LO1.1Ab
Students will know that…
 Given a function 𝑓, the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 is a real number 𝑅 if 𝑓(𝑥) can be
made arbitrarily close to 𝑅 by taking 𝑥 sufficiently close to 𝑐 (but not equal to 𝑐). If the
limit exists and is a real number, then the common notation is lim 𝑓(𝑥) = 𝑅.

𝑥→𝑐
The concept of a limit can be extended to include one-sided limits, limits at infinity, and
infinite limits.
Lim.C.2 Determine limits of functions
LO1.1C
LO1.1Ab
LO1.1B
Students will know that…
 Limits of sums, differences, products, quotients, and composite functions can be found
using the basic theorems of limits and algebraic rules
 The limit of a function may be found by using algebraic manipulation, alternate forms of
trigonometric functions (trig identities), or the squeeze theorem.
 A limit might not exist for some functions at particular values of 𝑥. Some ways that the
limit might not exist are if the function is unbounded, if the function is oscillating near this
value, or if the limit from the left does not equal the limit from the right.
0
∞
 Limits of the indeterminate forms 0 and ∞ may be evaluated using L’Hospital’s Rule.
 Numerical and graphical information can be used to estimate limits.
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Derivatives
Students will understand that:



The derivative of a function is defined as the limit of a difference quotient and can be determined using
a variety of strategies.
A function’s derivative, which is itself a function, can be used to understand the behavior of the function.
The derivative has multiple representations and applications including those that involve instantaneous
rates of change.
Deriv.B.1 Use derivatives to analyze properties of a function
LO2.2A
Students will know that…
 First and second derivatives of a function can provide information about the function
and its graph including intervals of increase or decrease, local (relative) and global
(absolute) extrema, intervals of upward or downward concavity, and points of inflection.
 Key features of functions and their derivatives can be identified and related to their
graphical, numerical, and analytical representations.
 Key features of the graphs of 𝑓, 𝑓 ′ , and 𝑓 ′′ are related to one another
Deriv.B.2 Recognize the connection between differentiability and continuity
LO2.2B
Students will know that…
 A continuous function may fail to be differentiable at a point in its domain
 If a function is differentiable at a point, then it is continuous at that point
Deriv.B.3 Interpret the meaning of a derivative within a problem
LO2.3A
Students will know that…
 The unit for 𝑓 ′ (𝑥) is the unit for 𝑓 divided by the unit for 𝑥.
 The derivative of a function can be interpreted as the instantaneous rate of change with
respect to its independent variable
Deriv.B.4 Solve problems involving the slope of a tangent line
LO2.3B
Students will know that…
 The derivative at a point is the slope of the line tangent to a graph at that point on the
graph.
 The tangent line is the graph of a locally linear approximation of the function near the
point of tangency.
Deriv.B.5 Solve problems involving optimization
LO2.3C
Students will know that…
 The derivative can be used to solve optimization problems, that is, finding a maximum or
minimum value of a function over a given interval.
 Parameters can be used to add flexibility to a model.
Deriv.B.6 Solve problems involving related rates
LO2.3C
Students will know that…
 The derivative can be used to solve related rates problems, that is, finding a rate at which
one quantity is changing by relating it to other quantities whose rates of change are
known.
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Deriv.B.7 Solve problems involving rectilinear motion
LO2.3C
Students will know that…
 The derivative can be used to solve rectilinear motion problems involving position, speed,
velocity, and acceleration.
Deriv.B.8 Verify solutions to differential equations
LO2.3E
Students will know that…
 Solutions to differential equations are functions or families of functions
 Derivatives can be used to verify that a function is a solution to a given differential
equation.
Deriv.B.9 Estimate solutions to differential equations
LO2.3F
Students will know that…
 Slope fields provide visual clues to the behavior of solutions to first order differential
equations
 For differential equations, Euler’s method provides a procedure for approximating a
solution or a point on a solution curve.
Deriv.C.1 Identify the derivative of a function as the limit of a difference quotient
LO2.1A
Students will know that…


𝑓(𝑎+ℎ)−𝑓(𝑎)
𝑓(𝑥)−𝑓(𝑎)
The difference quotients
and 𝑥−𝑎 express the average rate of change of a
ℎ
function over an interval
The instantaneous rate of change of a function at a point can be expressed by
lim
ℎ→0
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
or lim
𝑥→𝑎
𝑓(𝑥)−𝑓(𝑎)
,
𝑥−𝑎
provided that the limit exists. These are common forms of
the definition of the derivative and are denoted 𝑓 ′ (𝑎).
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
ℎ→0

The derivative of 𝑓 is the function whose value at 𝑥 is lim

exists
The derivative can be represented graphically, numerically, analytically, and verbally.
provided this limit
Deriv.C.2 Estimate derivatives
LO2.1B
Students will know that…
 The derivative at a point can be estimated from information given in tables or graphs
Deriv.C.3 Calculate explicit derivatives
LO2.1C
Students will know that…
 Direct application of the definition of the derivative can be used to find the derivative for
selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic
functions.
 Specific rules can be used to calculate derivatives for classes of functions, including
polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse
trigonometric.
 Sums differences products, and quotients of functions can be differentiated using
derivative rules.
 The chain rule provides a way to differentiate composite functions
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Deriv.C.4 Calculate implicit derivatives
LO2.1C
Students will know that…
 The chain rule is the basis for implicit differentiation.
 The chain rule can be used to find the derivative of an inverse function, provided that
derivative of that function exists.
Deriv.C.5 Express derivatives symbolically with correct notation
LO2.1A
LO2.1D
Students will know that…
𝑑𝑦
 For 𝑦 = 𝑓(𝑥), notations for the derivative include 𝑑𝑥 , 𝑓 ′ (𝑥), and 𝑦 ′ .
 Higher order derivatives are represented with a variety of notations. For 𝑦 = 𝑓(𝑥),
𝑑2 𝑦
notations for the second derivative include 𝑑𝑥 2 , 𝑓 ′′ (𝑥), and 𝑦 ′′ . Higher order derivatives
𝑑𝑛𝑦
can be denoted 𝑑𝑥 𝑛 or 𝑓 (𝑛) (𝑥).
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Integrals and the Fundamental Theorem of Calculus
Students will understand that:





Antidifferentiation is the inverse process of differentiation.
The definite integral of a function over an interval is the limit of a Riemann sum over that interval and can
be calculated using a variety of strategies.
The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and
integration.
The definite integral of a function over an interval is a mathematical tool with many interpretations and
applications involving accumulation.
Antidifferentation is an underlying concept involved in solving separable differential equations. Solving
separable differential equations involves determine a function or relation given its rate of change.
Int.B.1
Interpret the definite integral as the limit of a Riemann sum
LO3.2A(a)
Students will know that…
 A Riemann sum, which requires a partition of an interval 𝐼, is the sum of products, each
of which is the value of the function at a point in a subinterval multiplied by the length of
that subinterval of the partition.
Int.B.2
Express the limit of a Riemann sum in integral notation
LO3.2A(b)
Students will know that…
 The definite integral of a continuous function 𝑓 over the interval [𝑎, 𝑏], denoted by
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥, is the limit of Riemann sums as the widths of the subintervals approach 0
 The information in a definite integral can be translated into the limit of a related
Riemann sum, and the limit of a Rieman sum can be written as a definite integral..
Int.B.3
Analyze functions defined by an integral
LO3.3A
Students will know that…
 The definite integral can be used to define new functions; for example,
𝑥
2
𝑓(𝑥) = ∫0 𝑒 −𝑡 𝑑𝑡.


𝑑
𝑥
If 𝑓 is a continuous function on the interval [𝑎, 𝑏], then 𝑑𝑥 (∫𝑎 𝑓(𝑡) 𝑑𝑡) = 𝑓(𝑥), where 𝑥
is between 𝑎 and 𝑏.
Graphical, numerical, analytical, and verbal representations of a function 𝑓 provide
𝑥
information about the function 𝑔 defined as 𝑔(𝑥) = ∫𝑎 𝑓(𝑡) 𝑑𝑡.
Int.B.4
Interpret the meaning of a definite integral within a problem
LO3.4A
Students will know that…
 A function defined as an integral represents an accumulation of a rate of change
 The definite integral of the rate of change of a quantity over an interval gives the net
change of that quantity over that interval
 The limit of an approximating Riemann sum can be interpreted as a definite integral.
Int.B.5
Use the definite integral to solve problems in various contexts
LO3.4E
Students will know that…
 The definite integral can be used to express information about accumulation and net
change in many applied contexts
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Int.B.6
Analyze differential equations to obtain general and specific solutions.
LO3.5A
Students will know that…
 Antidifferentiation can be used to find specific solutions to differential equations with
given initial conditions, including applications to motion along a line, exponential growth
and decay.
 Some differential equations can be solved by separation of variables
 Solutions to differential equations may be subject to domain restrictions
𝑥
 The function 𝐹 defined by 𝐹(𝑥) = 𝑐 + ∫𝑎 𝑓(𝑡) 𝑑𝑡 is a general solution to the differential
𝑥
𝑑𝑦
equation 𝑑𝑥 = 𝑓(𝑥), and 𝐹(𝑥) = 𝑦0 + ∫𝑎 𝑓(𝑡) 𝑑𝑡 is a particular solution to the
𝑑𝑦
differential equation 𝑑𝑥 = 𝑓(𝑥) satisfying 𝐹(𝑎) = 𝑦0 .
Int.B.7
Interpret, create and solve differential equations from problems in context.
LO3.5B
Students will know that…
 The model for exponential growth and decay that arises from the statement “The rate of
𝑑𝑦
change of a quantity is proportional to the size of the quantity” is = 𝑘𝑦.
𝑑𝑡
Int.C.1
Recognize antiderivatives of basic functions
LO3.1A
Students will know that…
 An antiderivative of a function 𝑓 is a function 𝑔 whose derivative is 𝑓.
 Differentiation rules provide the foundation for finding antiderivatives.
Int.C.2
Approximate a definite integral
LO3.2B
Students will know that…
 Definite integrals can be approximated for functions that are represented graphically,
numerically, algebraically, and verbally.
 Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a
midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using
either uniform or nonuniform partitions.
Int.C.3
Calculate a definite integral using areas and properties of definite integrals
LO3.2C
Students will know that…
 In some cases, a definite integral can be evaluated by using geometry and the
connection between the definite integral and area
 Properties of definite integrals include the integral of a constant times a function, the
integral of the sum of two functions, reversal of limits of integration, and the integral of
a function over adjacent intervals.
 The definition of the definite integral may be extended to functions with removable or
jump discontinuities
Int.C.4
Calculate antiderivatives and evaluate definite integrals
LO3.3B
Students will know that…
𝑥
 The function defined by 𝐹(𝑥) = ∫𝑎 𝑓(𝑡) 𝑑𝑡 is an antiderivative of 𝑓.
 If 𝑓 is a continuous function on the interval [𝑎, 𝑏], and 𝐹 is an antiderivative of 𝑓, then
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎).
 Many functions do not have closed form antiderivatives
NCSSM – AP Calculus AB – 2014-2015 Learning Standards
Int.C.5
Use advanced techniques of integration
LO3.3B
Students will know that…
 Techniques for finding antiderivatives include algebraic manipulation such as long
division and completing the square, and substitution of variables (u-substitution)
Int.C.6
Apply definite integrals to problems involving the average value of a function
LO3.4B
Students will know that…
𝑏
1
 The average value of a function 𝑓 over an interval [𝑎, 𝑏] is 𝑏−𝑎 ∫𝑎 𝑓(𝑥) 𝑑𝑥.
Int.C.7
Apply definite integrals to problems involving motion
LO3.4C
Students will know that…
 For a particle in rectilinear motion over an interval of time, the definite integral of
velocity represents the particle’s displacement over the interval of time, and the definite
integral of speed represents the particle’s total distance traveled over the interval of
time.
Int.C.8
Apply definite integrals to problems involving area, volume, and length of a curve
LO3.4D
Students will know that…
 Areas of certain regions in the plane can be calculated with definite integrals.
 Volumes of solids with known cross sections, including discs and washers, can be
calculated with definite integrals.
 The length of a planar curve defined by a function or by a parametrically defined curve
can be calculated using a definite integral.